Theorem elima 5087

Description: Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)

Hypothesis

Ref	Expression
elima.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
elima	⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem elima

Step	Hyp	Ref	Expression
1		elima.1	. 2 ⊢ 𝐴 ∈ V
2		elimag 5086	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴)

Syntax hints:   ↔ wb 105   ∈ wcel 2202  ∃wrex 2512  Vcvv 2803   class class class wbr 4093   “ cima 4734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744

This theorem is referenced by:  elima2  5088  rninxp  5187  imaco  5249  isarep1  5423  funimass4  5705